Solution of non-linear boundary integral equations in complex geometries with auxiliary integral subtraction

摘要

The boundary integral equation that results from the application of the reciprocity theorem to non-linear or non-homogeneous differential equations generally contains a domain integral. While methods exist for the meshless evaluation of these integrals, mesh-based domain integration is generally more accurate and can be performed more quickly with the application of fast multipole methods. However, polygonalization of complex multiply-connected geometries can become a costly task, especially in three-dimensional analyses. In this paper, a method that allows a mesh-based integration in complex domains, while retaining a simple mesh structure, is described. Although the technique is intended for the numerical solution of more complex differential equations, such as the Navier-Stokes equations, for simplicity the method is applied to the solution of a Poisson equation, in domains of varying complexity. It is shown that the error introduced by the auxiliary domain subtraction method is comparable to the discretization error, while the complexity of the mesh is significantly reduced. The behaviour of the error in the boundary solution observed with the application of the new method is analogous to the behaviour observed with conventional cell-based domain integration.